Cédric Galusinski

Uniform exponential attractors for a singularly perturbed damped wave equation

Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.

On stability condition for bifluid flows with surface tension: Application to microfluidics

Models for incompressible immiscible bifluid flows with surface tension are here considered. Since Brackbill et al. [J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (1992) 335-354] introduced the Continuum Surface Force (CSF) method, many methods involved in interface tracking or capturing are based on this reference work. Particularly, the surface tension term is discretized explicitly and therefore, a stability condition is induced on the computational time step. This constraint on the time step allows the containment of the amplification of capillary waves along the interface and puts more emphasis on the terms linked with the density in the Navier-Stokes equation (i.e. unsteady and inertia terms) rather than on the viscous terms. Indeed, the viscosity does not appear, as a parameter, in this stability condition. We propose a new stability condition which takes into account all fluid characteristics (density and viscosity) and for which we present a theoretical estimation. We detail the analysis which is based on a perturbation study - with capillary wave - for which we use energy estimate on the induced perturbed velocity. We validate our analysis and algorithms with numerical simulations of microfluidic flows using a Level Set method, namely the exploration of different mixing dynamics inside microdroplets.

Level-set, penalization and cartesian meshes: A paradigm for inverse problems and optimal design

The aim of this work is to combine penalization and level-set methods to solve inverse or shape optimization problems on uniform cartesian meshes. Penalization is a method to impose boundary conditions avoiding the use of body-fitted grids, whereas level-sets allow a natural non-parametric description of the geometries to be optimized. In this way, the optimization problem is set in a larger design space compared to classical parametric representation of the geometries, and, moreover, there is no need of remeshing at each optimization step. Special care is devoted to the solution of the governing equations in the vicinity of the penalized regions and a method is introduced to increase the accuracy of the discretization. Another essential feature of the optimization technique proposed is the shape gradient preconditioning. This aspect turns out to be crucial since the problem is infinite dimensional in the limit of grid resolution. Examples pertaining to model inverse problems and to shape design for Stokes flows are discussed, demonstrating the effectiveness of this approach.

Waves in ferromagnetic media.

ABSTRACT It is shown that small perturbations of equilibrium states in ferromagnetic media give rise to standing and traveling waves that are stable for long times. The evolution of the wave profiles is governed by semilinear heat equations. The mathematical model underlying these results consists of the Landau–Lifshitz equation for the magnetization vector and Maxwells equations for the electromagnetic field variables. The model belongs to a general class of hyperbolic equations for vector-valued functions, whose asymptotic properties are analyzed rigorously. The results are illustrated with numerical examples.

Simulations of 3D Dynamics of Microdroplets: A Comparison of Rectangular and Cylindrical Channels

In this paper, several numerical simulations of diphasic flows in microchannels are presented. The flow in both cylindrical and rectangular channels is considered. The aim is to compute the shape of the droplets and the velocity fields inside and outside the droplets and to quantify the influence of the geometry. The Level Set method is used to follow the interface between the fluids.